Dimension vectors in regular components over wild Kronecker quivers

TL;DR

This paper investigates the relationship between dimension vectors and quasi-lengths of indecomposable regular representations in wild Kronecker quivers, addressing bounds and coincidences across regular components.

Contribution

It provides new insights into the structure of regular components over wild Kronecker quivers, including relationships between dimension vectors and quasi-lengths, bounds on indecomposables, and conditions for vector set coincidences.

Findings

Established the relationship between dimension vectors and quasi-lengths.

Derived bounds on the number of indecomposables with fixed length.

Characterized when dimension vector sets coincide across components.

Abstract

Let $\mathcal{K}_n$ be the so-called wild Kronecker quiver, i.e., a quiver with one source and one sink and $n\geq 3$ arrows from the source to the sink. The following problems will be studied for an arbitrary regular component $\mathcal{C}$ of the Auslander-Reiten quiver: (1) What is the relationship between dimension vectors and quasi-lengths of the indecomposable regular representations in $\mathcal{C}$? (2) For a given natural number $d$, is there an upper bound of the number of indecomposable representations in $\mathcal{C}$ with the same length $d$? (3) When do the sets of the dimension vectors of indecomposable representations in different regular components coincide?